Just trying to think why it would pee when in the lake and not in the tank....
My first thought is that it's not priming for some reason.
On some outboards, there are water inlets at the front of the gearcase that must be taped closed in order to get a proper seal for using muffs. I'm not sure if that applies to yours or not but something you may want to check for.
Besides that, muffs are very finicky anyway about getting a good seal and, if they don't fit right they won't work. That could cause damage to the pump and or unintentional overheat. It's why I prefer not to use muffs. Just my preference.
As far as priming and pumping in the lake and not in the tank, ....do you mean when the boat is moving or stationary?
If it's different when stationary, I tend to think it's likely that the outboard sits deeper in the water when in the lake than when in your drum.
You might need to modify the drum so that the lower end of the outboard is submerged at least as much in the tank as it is when in the lake.
A 30 gallon, plastic trashcan can be a good choice for use as a test tank because it's easier to cut and shape as well as being flexible and much lighter than the drum.
If it only pees when the boat is moving then that's a new one on me.
This outboard has a pick-up tube that should be closely inspected for even a tiny hole or small crack as that could affect the pump's ability to pick up prime.
See item 3 in the link.
It's also a good idea to remove the intake screens (items 13 & 14) and ensure that there's no weeds or other items in the cavity that might interfere with the pump picking up the water.
I see this outboard extension has a drain plug...item 8.
I don't know how that might effect priming but I would check it out. If the plug is missing or is badly deteriorated, it might be having a negative effect. I don't know.
Also if the speedometer tube isn't being used (items 17, 18 & 59) you might want to try plugging that off to see if it makes a difference. I'm not sure it should but I don't know.